C3 Jurv- \3 (ncpa<-c-A po$€r) t.ttr*1. Given that

3xa-2x3-5xz-4 1 . dx+e

t__;rf

(4)

3r-* _-if- +}f i -8* t

TI-rrf +E:c -Lg i. {rLv

I

Given thatf(x):lnx, x>0

sketch on separate axes the graphs of

(i) y: f(x),

(ii) /: lf(n) 1,

(iii) y:-f(x-4).

Show, on each diagram, the point where the graph meets or crosses the x-axis,In each case, state the equation of the asymptote.

$ttx-)(7)

asU'\^P+ote-

a-= O

I

I

I

I

I

I

+'l

I

:

aYrnp\ote-

aSSrnOtofre-

*=O

U'te(*;l

g---(r+)

w*

3. Given that

2cos(x + 50)" : sin(x + 40)'

(a) Show, without using a calculatot, that

tan xo : l tan 4o'1J

(b) Hence solve, for 0 ( 0 <360,

2cos(20 + 50)" : sin(20 + 40)'

giving your answers to 1 decimal place'

O.CpS:--Cos-5O -e.Sqn,:r-S,1n 50 :- Srn>9(os QO- + CosI- Srn\O

e O,S-rv.reO --' e.\rrnX-,Co5 *O :- hr^nX (,s.4D +Srn

-- kr,*n)L + trrnYO

tt^-n 1=, tt"".SP

f(x):25x2e2'-76' re lR

(a) Using calculus, find the exact coordinates of the turning points on the

equation y:f(x)

(b) Show that the equation f(x) : 0 can be written as r : *1"*

The equation f(r) : 0 has a root a, where a: 0.5 to 1 decimal place'

(c) Starting with xo : 0.5, use the iteration formula

4Xn t: je-r'ri

of x, xrattd x, giving your answers to 3 dec

mate for a to 2 decimal places, and justify y

e))

curve with

(s)

(1)

ues

stin

to calculate the valt

(d) Give an accurate es

c)- L,o- i-O S- ,[X-r_;= O

' _!t55

d) +(-o'kgs): :O: *q < O_flCo.rr1s)'_ t o " trl__?o_

.7rz : O*frL- *tl-s -., o: tt89

5. Given that

x : sec2 3Y, 0 'Y 'L6

d"r(a) find ; in terms ofY.dy (2)

(b) Hence show that

dv- 1

d. - 6{.i,(4)

dzv(c) Find an expression for |f in terms of x. Give your answer in its simplest form.

(4)

cA >c*--r-( sec.st):

= l(Se-3",r) x 3Sec3,,tn^3\\-

x-:Scc1b-,;6-= bef3**,\a-r:Yf1_s-

;-t ctqA! -- Il.fioI-{I--\

(:-il =- [qp133

1

clx_ _-_ a(:c-r) a _utr;,_\_:f-. vr, 3o..)t

-'.4b--,'3

--

I

{oc-Da-&36 (x--\)

rl.:bCx.-rXr-t\L - 3r-,

--9--j6* Ct-t)T' 3b*(-1{;1'

e-3x=) tffit>zr> _b

6. Find algebraically the exact solutions to the equations

(a) 1n(4 - 2x) + ln(9- 3x) : 2ln(x + l), -l < x < 2

(b) 2'e"*' : 10

a+1nbGive your answer to (b) in the form

" + tn d where a. b, c and d are integers.

;;_W{L*.,Jai .,;I;k{iiirf I ",

e-5t':j2:r+3€ -O -> (s,c-+Xr:S) "s

rr) ,(*d'+r ) = ln \o

=) lnZ" + lr.re-H*\ = \rr \O

--=i---r( i;2) 3 -\ +htO

(s)

(s)

3+\n2-

7. The function f has domain -2 ( x ( 6 and is linear from (-2, 10) to (2, 0) and from (2' 0)

to(6,4).Asketchofthegraphofy:f(x)isshowninFigurel'

Figure 1

(a) Write down the range of f.

(b) Find ff(0).

The function g is defined bY

4+3xg:x-> 5---r'

(c) Find g-1(x)

(d) Solve the equation gf(x) : 16

(1)

(2)

(3)

(s)

xelR, x+5

a) esj_.tOb)

' G;$-"-

--- -Gto)is- i *--

4 -,--L- i+ '(sro;

c) Jc = t*3:r5=-S

5)e.-f,J = 4,+33 --

c) "\{t-1 = t6 =) t - 3+(=) =o - 5-*(r) t6

l;)

(+- ,+ x-= (> ,

8.

24mFigure 2

Kate crosses a road, of constant width 7 m, rn order to take a photograph of a marathon

runner, John, approaching at 3 m s-1.

Kate is 24 m aiead of John when she starts to cross the road from the fixed point ,4.

John passes her as she reaches the other side of the road at a variable point B, as shown in

Figure 2.

Kolte,s speed is Vms-l and she moves in a straight line, which makes an angle 0,

O < 0 < 150o, with the edge of the road, as shown in Figure 2'

You may assume that V is given by the formula

2l 0<0<150'V-24sin0 + 7 cos?'

(a) Express 24sin0 + 7cos7 in the form Rcos(0 - o), where R and a are constants and

where ft > 0 and 0 I a r-90o, giving the value of a to 2 decimalplaces.(3)

Given that 0 varies,

(b) frnd the minimum value of Z

Given that Kate's speed has the value found in part (b),

(c) find the drstance AB.

Given instead that Kate's speed is 1.68 ffi s-1,

(d) find the two possible values of the angle 0, given that 0 < P < 150o

(2)

(3)

(6)

ro) K tor($ -o() = IGu,dAi* * f sr6i,Ja1gff -*- zQsyd

trnrn c( = ?\, + &- *3.+\?-- L5

-=--t-68

-E-*--V*.,,tfhax (2-SCos{s:13:+q) 2.S,

--.+- -C-o-sIo -ei ?,$) :,L\ - ;-OSzsx(-68',